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The method is quite general as regards to loss function and underlying distribution. It has proven useful in unrestricted as well as restricted parameter spaces. In particular, one of its first uses was to produce an estimator dominating the James-Stein estimator of a multivariate normal mean under squared error loss. In Section 4, following a discussion on questions of admissibility concerning estimators that take values on the boundary of a restricted parameter space, we investigate a technique of Moors (1985) which is useful in constructing improvements to such estimators under squared error loss.

Since C was an arbitrary λ-proper set, the chain W is locally-ν-recurrent. Acknowledgment Many thanks to Jim Hobert, Tiefeng Jiang and Galin Jones for their valuable comments. Also special thanks to Anirban Das Gupta for his efforts on this Festschrift for Herman Rubin and his many comments on this contribution. References [1] Blackwell, D. (1951). On the translation parameter problem for discrete variables. Ann. Math. Statist. 22, 105–110. MR43418 [2] Blyth, C. R. (1951). On minimax statistical procedures and their admissibility.

6) Because g(y) ∈ [0, 1], (g(y) − g(z))2 R(dy|z)λ(dz) ∆(g) = = 2 ≤ 2 g 2 (y)λ(dy) − g(y)g(z)R(dy|z)λ(dz) g 2 (y)λ(dy) ≤ 2λ B ∩ AC . 6) yields 1 1 1 1 H 2 (B) ≤ ∆ 2 (h) + 2 2 λ 2 B ∩ Ac . 3) holds. This completes the proof. 4) follows. For h1 and h2 in L2 (λ), consider the symmetric bilinear form < h1 , h2 >= h1 (y)h2 (y)λ(dy) − h1 (y)h2 (z)R(dy|z)λ(dz). That < ·, · > is non-negative definite is a consequence of the symmetry of R and the Cauchy–Schwarz inequality: 2 h1 (y)h1 (z)R(dy|z)λ(dz) ≤ h21 (y)R(dy|z)λ(dz) · 2 = h21 (y)λ(dy) .

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